I've copied an entire WP multisite from one server to another. After some realignment, I am left with just one issue with Custompress. The Custom Taxonomies and Custom Fields are both correctly populated, but the Custom Post Types tab is displaying empty. There are 3 custom post types that were created. I searched the database and can see that all the individual posts are there with the affiliated post type. I can also see all of the wp_options copied over exactly.
I noticed that someone else wrote about using BackupBuddy, but I'd rather not start over with the move.